Article
Developing an Improved Parameter Estimation
Method for the Segmented Taper Equation through
Combination of Constrained Two-Dimensional
Optimum Seeking and Least Square Regression
Lifeng Pang 1 , Yongpeng Ma 2 , Ram P. Sharma 3 , Shawn Rice 4 , Xinyu Song 5 and Liyong Fu 1,6, *
1
2
3
4
5
6
*
Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry,
Beijing 100091, China; plf619@ifrit.ac.cn
Key Laboratory for Plant Diversity and Biogeography of East Asia, Kunming Institute of Botany,
Chinese Academy of Sciences, Kunming 650201, Yunnan, China; mayongpeng@mail.kib.ac.cn
Faculty of Forestry and Wood Sciences, Czech University of Life Sciences Prague, Prague 6, Czech Republic;
ramsharm1@gmail.com
Penn State Hershey Cancer Institute, Penn State College of Medicine, Hershey, PA 17033, USA;
srice@hmc.psu.edu
College of Computer and Information Techniques, Xinyang Normal University, Xinyang 464000, Henan,
China; xysong88@163.com
Center for Statistical Genetics, Pennsylvania State University, Loc T3436, Mailcode CH69,
500 University Drive, Hershey, PA 17033, USA
Correspondence: fuliyong840909@163.com; Tel.: +86-10-6288-9179; Fax: +86-10-6288-8315
Academic Editors: Jean-Claude Ruel and Eric J. Jokela
Received: 21 July 2016; Accepted: 27 August 2016; Published: 31 August 2016
Abstract: The segmented taper equation has great flexibility and is widely applied in exiting taper
systems. The unconstrained least square regression (ULSR) was generally used to estimate parameters
in previous applications of the segmented taper equations. The joint point parameters estimated with
ULSR may fall outside the feasible region, which leads to the results of the segmented taper equation
being uncertain and meaningless. In this study, a combined method of constrained two-dimensional
optimum seeking and least square regression (CTOS & LSR) was proposed as an improved method
to estimate the parameters in the segmented taper equation. The CTOS & LSR was compared
with ULSR for both individual tree-level equation and the population average-level equation
using data from three tropical precious tree species (Castanopsis hystrix, Erythrophleum fordii, and
Tectona grandis) in the southwest of China. The differences between CTOS & LSR and ULSR were
found to be significant. The segmented taper equation estimated using CTOS & LSR resulted in not
only increased prediction accuracy, but also guaranteed the parameter estimates in a more meaningful
way. It is thus recommended that the combined method of constrained two-dimensional optimum
seeking and least square regression should be a preferred choice for this application. The computation
procedures required for this method is presented in the article.
Keywords: segmented taper equation; unconstrained least square regression; constrained
two-dimensional optimum seeking; parameter estimation; precious tree species
1. Introduction
The mathematical function (or equation) describing the variation of the tree diameter at any point
of the stem with the distance from the tree top is known as the stem taper equation [1]. Using this
equation, one may calculate stem diameter at any arbitrary height and conversely, calculate tree height
for any arbitrary stem diameter. Consequently, stem volume can be calculated for any log specification,
Forests 2016, 7, 194; doi:10.3390/f7090194
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Forests 2016, 7, 194
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and a volume equation can be developed for classified product dimensions [2]. In many countries,
especially the United States and Canada, stem taper equations have been commonly used to replace
volume tables and volume equations for estimating stem volume, merchantable volume ratio, 3-D
reconstruction of stem and simulation, and optimization of bucking [3–5]. Therefore, developing
a stem taper equation with high predictive precision has always been a matter of interest to forest
managers and forest engineers as the taper equation provides technical support to forest management.
Numerous forms of the taper equation have been developed over the past century ranging
from simple to complex. These taper equations can be divided into three categories: simple taper
equation [1,6–12], segmented taper equation [13–17], and variable-exponent taper equation [3,18–22].
Simple taper equations are easy to fit, and can be used to calculate the stem volume applying the
integration method. Relatively simple taper functions can effectively describe the general taper of
the trees. However, they lack the ability to describe the entire stem accurately. They may provide
reasonable estimates in a mid-portion of the stem, but usually are less accurate in estimating the
profile in the butt or upper stem segments [3,13,18,23]. As compared to the simple taper equation, the
segmented taper equation exhibits a higher precision of diameter estimation and also could be used
to describe the entire stem profile more efficiently [13,24,25]. The variable-exponent taper equation
also has a high estimation precision, but it cannot be used to estimate volume by integration without
the aid of a computer program [3,17,26]. Overall, the segmented taper equation has greater flexibility
than other taper equations and therefore has frequently been applied along with different modelling
approaches (e.g., traditional least square regression, seemingly unrelated regression and nonlinear
mixed-effects model) [14,17,24,25].
The Max and Burkhart [13] taper equation is one of the most typical segmented taper equations to
describe tree stem taper [14,25,27]. This equation consists of three different quadratic functions joined
at two joint points, and therefore is flexible enough to describe complex and irregular stem profiles.
Many researchers have used this equation for the prediction of tree taper [15,17,24,25,28–32]. Two joint
point (upper and lower) parameters (the feasible region of two parameters lies between 0 and 1) in the
Max and Burkhart [13] taper equation were usually estimated by applying a traditional unconstrained
least square regression (ULSR) method and subjective experience (SE) method [13,17,25,33]. However,
in a practical application, it has been found that the estimates of the two joint point parameters obtained
from ULSR sometimes fall outside the feasible region of the corresponding estimated parameters,
which makes the results meaningless [34]. The study involved the application of a simplex method
with constraint to estimate parameters of the Max and Burkhart [13] taper equation. However, this
failed to converge and was found very slow computationally. In addition, the study also revealed that
the estimates of the two joint point parameters obtained by ULSR varied with the starting values of
the parameters. Therefore, the developed Max and Burkhart [13] taper equation based on ULSR is
subject to a great uncertainty. The SE method has a great uncertainty in estimation of the join point
parameters due to unpredictable man-induced factors and inevitable measurement errors [35]. These
uncertainties directly influence the estimation precision and practicability of the taper equations.
The objective of this study was thus to propose a combined method of constrained
two-dimensional optimum seeking and least square regression (CTOS & LSR) as an improved
method to estimate parameters of the segmented taper equation based on Hua’s optimum seeking
theory and golden section search approach [36]. The data from three tropical precious tree species
Castanopsis hystrix Miq. (C. hystrix), Erythrophleum fordii Oliv. (E. fordii), and Tectona grandis L.F.
(T. grandis) in southwest China were used to evaluate the methods. Also, CTOS & LSR was compared
with the commonly used ULSR approach by means of developing the Max and Burkhart [13] taper
equation at both individual tree-levels and population average-levels.
Forests 2016, 7, 194
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2. Materials and Methods
2.1. Data
We used data from a total of 30 square-shaped permanent sample plots (PSPs) established
Forests 2016, 7, 194 3 of 20 in 2014 across three pure even-aged C. hystrix, E. fordii, and T. grandis plantations (each tree species
with 10 PSPs)
located in the Experimental Center of Tropical Forestry, Chinese Academy of Forestry
We used data from a total of 30 square‐shaped permanent sample plots (PSPs) established in 2014 across three pure even‐aged C. hystrix, E. fordii, and T. grandis plantations (each tree species with Sciences
(Figure 1). C. hystrix, E. fordii, and T. grandis are tropical precious tree species in southern
10 The
PSPs) located the Experimental Center of Tropical Chinese Academy of Forestry China.
data
werein collected
by the Research
Institute
ofForestry, Forest Resources
Information
Techniques,
Sciences (Figure 1). C. hystrix, E. fordii, and T. grandis are tropical precious tree species in southern Chinese Academy of Forestry. Stands were selected on the basis of the following criteria: (1) pure
China. The data were collected by the Research Institute of Forest Resources Information Techniques, species; (2) fully stocked; (3) even-aged; and (4) minimum or no disturbance. Sample plots were
Chinese Academy of Forestry. Stands were selected on the basis of the following criteria: (1) pure avoided near major roads because road construction might have altered soil drainage and affected tree
species; (2) fully stocked; (3) even‐aged; and (4) minimum or no disturbance. Sample plots were growth.
Once a stand was located, a 600 m2 permanent sample plot was established. Four trees per
avoided near major roads because road construction might have altered soil drainage and affected sample
plot, each representing dominant height,2 permanent sample plot was established. Four trees codominant height, average height, and suppressed
tree growth. Once a stand was located, a 600 m
heightper were
selected.
These
trees
were
felled,
and
cross-sectional
slices
of stem
(disks)
were and obtained
sample plot, each representing dominant height, codominant height, average height, suppressed height were selected. These trees were felled, and cross‐sectional slices of stem (disks) from 0.33
m, 0.67 m, 1.3 m, and every 1.3 m interval thereafter until an outside bark diameter of 7.0 cm
were obtained from 0.33 m, 0.67 m, 1.3 m, and every 1.3 m interval thereafter until an outside bark was reached.
Disks were then taken every 20 cm between 7 cm and 4 cm outside bark diameter. Many
diameter of 7.0 cm was reached. Disks were then taken every 20 cm between 7 cm and 4 cm outside tree characteristics such as outside bark diameter at breast height (D), total height (H), height to live
diameter. Many tree characteristics such as outside bark diameter at breast height (D), total crownbark base,
and crown width, were measured for each sample tree. Summary statistics of D and H
height (H), height to live crown base, and crown width, were measured for each sample tree. data are listed in Table 1. We applied two approaches for fitting and testing the Max and Burkhart [13]
Summary statistics of D and H data are listed in Table 1. We applied two approaches for fitting and tapertesting the Max and Burkhart [13] taper equations; (i) at individual tree‐level using sub‐dataset of equations; (i) at individual tree-level using sub-dataset of nine dominant and codominant
trees (three
trees from each tree species) (Figure 2); (ii) at population average-level using full dataset
nine dominant and codominant trees (three trees from each tree species) (Figure 2); (ii) at population of 120average‐level using full dataset of 120 trees (Table 1). trees (Table 1).
Figure 1. Location of study area: Experimental Center of Tropical Forestry, Chinese Academy of Figure 1. Location of study area: Experimental Center of Tropical Forestry, Chinese Academy of
Forestry Sciences and spatial distribution of 30 sample plots. Forestry Sciences and spatial distribution of 30 sample plots.
Table 1. Summary statistics of full dataset. Table 1. Summary statistics of full dataset.
Tree Species Variable No. Mean Max Min SD CV D (cm) 40 22.12 37.55 8.70 7.40 0.33 Tree Species
Variable
No.
Mean
Max
Min
SD
CV
C. hystrix H (m) 40 21.43 30.50 10.10 5.21 0.24 D (cm) D (cm)
40
8.70 4.52 7.40
0.33
4022.1221.15 37.55
29.50 10.05
0.21 C. hystrix
E. fordii H (m)
40
21.43
30.50
10.10
5.21
0.24
H (m) 40 18.64 22.40 11.80 2.71 0.15 D (cm) D (cm) 40
10.05 5.49 4.52
0.21
40 21.1525.93 29.50
37.95 12.70 0.21 E. fordii T. grandi H (m) H(m) 40
18.64
22.40
11.80
2.71
0.15
40 21.16 27.40 9.30 3.52 0.17 D (cm)
40
25.93
37.95
12.70
5.49
0.21
D, diameter at breast height; H, total tree height; SD, standard deviation; CV, coefficient of variation. T. grandi
H(m)
40
21.16
27.40
9.30
3.52
0.17
D, diameter at breast height; H, total tree height; SD, standard deviation; CV, coefficient of variation.
Forests
2016, 7, 194
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C. hystrix 1
C. hystrix 2
C. hystrix 3
E. fordii 1
E. fordii 2
E. fordii 3
T. grandis 1
T. grandis 2
T. grandis 3
20
h (m)
15
10
5
0
0
10
20
30
40
50
60
d (cm)
Figure
2. Observed height (h) above the ground-outside bark diameter (d) curves for nine dominant or
Figure 2. Observed height (h) above the ground‐outside bark diameter (d) curves for nine dominant codominant
trees.
or codominant trees. 2.2.2.2. Taper Equation Taper Equation
The Max and Burkhart [13] taper equation is: The
Max and Burkhart [13] taper equation is:
y   ( x  1)   ( x2  1)   (  x)2 I   (  x)2 I   y = β1 ( 1x − 1) + β22( x2 − 1) + β3 3 (α1 1 − x )21I1 +4β4 (2α2 − x )22 I2 + ε
(1) (1)
where y  d / D , x  h / H , h is the height above ground to measurement point (m), d is the where y = d2 /D2 , x = h/H, h is the height above ground to measurement point (m), d is the outside
outside bark diameter at height h above ground (cm), H is the total tree height (m), D is the outside bark diameter at height h above ground (cm), H is the total tree height (m), D is the outside bark
bark diameter at breast height (cm), 1 to 4 , 1 ,  2 are parameters, I1 and I 2 are bole portion diameter at breast height (cm), β1 to β4 , α1 , α2 are parameters, I1 and I2 are bole portion indicator
indicator variables, and  is an error term with zero expectation. In Equation (1), 1 and  2 are variables, and ε is an error term with zero expectation. In Equation (1), α1 and α2 are upper
and lower
upper and lower joint point parameters, respectively, while I1  1 if 1  x and zero, otherwise, and joint point parameters, respectively, while I1 = 1 if α1 ≥ x and
zero, otherwise, and I2 = 1 if α2 ≥ x
 1 if  2  x and zero, otherwise. The value of 1 indicates relative height where shape of the bole andI 2zero,
otherwise. The value of α1 indicates relative height where shape of the bole changes from
changes from paraboloid of the middle bole portion to conical of the top bole portion, and that of 2 paraboloid
of the middle bole portion to conical of the top bole portion, and that of α2 indicates where
indicates where shape changes from neiloid of the butt portion to paraboloid of the middle portion shape
changes from neiloid of the butt portion to paraboloid of the middle portion of the tree bole.
of the tree bole. The methods for parameter estimation of Equation (1) evolved from simple traditional least
The methods for parameter estimation of Equation (1) evolved from simple traditional least square square regression
and seemingly unrelated regression to nonlinear mixed-effects model. An implicit
regression and seemingly unrelated regression to nonlinear mixed‐effects model. An implicit assumption shared by these methods is that the joint point parameters α1 and α2 are estimated without
assumption shared by these methods is that the joint point parameters 1 and  2 are estimated any constraint, which is in conflict with the condition of the feasible region
(i.e., 0 < α1 < α2 < 1).
without any constraint, which is in conflict with the condition of the feasible region (i.e., 0  1   2  1
Therefore,
we applied a constrained two-dimensional optimum seeking method (CTOS) based on the
). Therefore, we applied a constrained two‐dimensional optimum seeking method (CTOS) based on residual
sum of squares minimum theory to solve this problem. The corresponding objective function
the residual the
sum of squares isminimum for estimating
parameters
given bytheory to solve this problem. The corresponding objective 2
2
function for estimating the parameters is given by n
!
 f (θ) =n ∑ |y − 2ŷ|2
θ̂ =θˆ (β̂(1
,ˆβ̂,2
arg min
ˆ, β̂,3ˆ, β̂, 4ˆ, α̂, 1ˆ, α̂, 
2ˆ) =
)
arg
min

f ()  | y - yˆ |  1
2
3
4
1
2
0<0α
<α <1
i =1
11  221 
i=1

(2) (2)
ˆ , β̂ˆ , ˆβ̂, 
where,
θ̂ =θˆ (β̂(1ˆ, β̂
,ˆα̂,1ˆ, α̂)2is the estimates of equation parameter vector θ, ) is the estimates of equation parameter vector θ,
f ( is the residual θ) is the residual
where, f (θ)
1 , 22 , 33, 4 4
1
2
sum
of squares; y and
ŷ are
observed and estimated values, respectively.
ŷ are observed and estimated values, respectively. sum of squares; y and 2.3. CTOS & LSR Method
2.3. CTOS & LSR Method To solve for θ̂ in Equation (2), the CTOS & LSR method based on the optimum seeking method was
proposed in this study. The optimum seeking method was first proposed by American mathematician
Forests 2016, 7, 194 5 of 20 θ̂ in Equation (2), the CTOS & LSR method based on the optimum seeking method To solve for Forests 2016,
7, 194
5 of 20
was proposed in this study. The optimum seeking method was first proposed by American mathematician Jack Keifer in the 1950s when studying the single‐factor optimum seeking methods as inthe section method the
(also called the 0.618 method). The methods
purpose such
of the Jacksuch Keifer
thegolden 1950s when
studying
single-factor
optimum
seeking
as optimum the golden
seeking method to reduce the number The
of experiments to rapidly seeking
find out the optimal section
method
(alsowas called
the 0.618
method).
purpose of and the optimum
method
was to
experimental points by using mathematical principles in the production and scientific research [37]. reduce the number of experiments and to rapidly find out the optimal experimental points by using
The CTOS principles
method belongs to multivariate optimization approach, in which effects of two to
mathematical
in the production
and scientific
research
[37]. The
CTOSthe method
belongs
variables or factors on the objective function are considered, and an optimal point is sought under a multivariate
optimization approach, in which the effects of two variables or factors on the objective
given constraint condition. The general idea of the CTOS method was as follows (see Figure 3): (I) function are considered, and an optimal point is sought under a given constraint condition. The
The optimal point was first sought on a midline (e.g., red line); (II) Suppose the optimum was taken at general
idea of the CTOS method was as follows (see Figure 3): (I) The optimal point was first sought
point P1 , and a horizontal line passing through P1 is drawn (green line); (III) Then the optimum was on a midline (e.g., red line); (II) Suppose the optimum was taken at point P1 , and a horizontal line
sought on this (green line); Suppose the optimum occurs on
at this
point P2 , and an passing
through
P1horizontal is drawn line (green
line);
(III)(IV) Then
the optimum
was sought
horizontal
line
optimum was found on the vertical line (yellow line) passing through ; (V) The optimum was set P
2
(green line); (IV) Suppose the optimum occurs at point P2 , and an optimum
was found on the vertical
P3 ; (VI) The above procedures were repeated several times. The shadow indicated that this line at point (yellow line)
passing through P2 ; (V) The optimum was set at point P3 ; (VI) The above procedures
weresection was removed from the feasible region. Therefore, the feasible region was gradually narrowed repeated several times. The shadow indicated that this section was removed from the feasible
until the final optimal point was found. For CTOS & LSR, the joint point parameters  2 were region.
Therefore, the feasible region was gradually narrowed until the final optimal point
was
found.
1 and estimated iteratively in the CTOS method, then first
other estimated
parameters such as 
For first CTOS
& LSR, the
joint point
parameters
α1 andand α2 were
iteratively
in1 to theCTOS
4 in LSR. All of these given
parameters Equation (1) were estimated given ̂1 and method,
and then
other
parameters
such
as β1̂ 2to applying β4 in Equation
(1)estimates were estimated
α̂1 andin α̂2
applying
LSR.(1) Allin estimates
of these
parameters
in Equation
(1) the in each
iteration
finally
used to
Equation each iteration were finally used to compute values of the were
objective function compute
the values of the objective function (residual sum of squares).
(residual sum of squares). b2
b2
P1
P1
P2
P3
P3
a2
a1
a1  b1
2
P2
b1
a2
a1
a1  b1
2
b1
Figure 3. The general idea of the constrained two‐dimensional optimum seeking method, a
1 and b1 Figure
3. The general idea of the constrained two-dimensional optimum seeking method, a1 and
b1 are
are the lower and upper limits for the feasible region of the first parameter, respectively; a
2 and b2 are the lower and upper limits for the feasible region of the first parameter, respectively; a2 and b2 are the
the lower and upper limits for the feasible region of the second parameter, respectively, P1, P2, and P3 lower and upper limits for the feasible region of the second parameter, respectively, P1 , P2 , and P3 are
are corresponding optimal points for different iterative steps. corresponding optimal points for different iterative steps.
The golden section method which is the foundation of many optimization algorithms was The golden section method which is the foundation of many optimization algorithms was applied
applied in this study to improve the computational efficiency of CTOS & LSR [37]. Two straight lines in this
studyiteration to improve
the
computational
efficiency
CTOS
LSR [37].
Two straight
in each
in each were drawn at 0.382 and 0.618 of of
the value &ranges of variable 1 and lines
variable 2, iteration
were drawn at 0.382 and 0.618 of the value ranges of variable 1 and variable 2, respectively.
respectively. The intersections of these straight lines were called interior points indicated by G1 , G2 , G3
The , and intersections
of these straight lines
were called interior points indicated by G1 , Gf 2(G
, 1G) 3, , and
G4 , which formed a rectangle f (G2 )G4 ,
U (see Figure 4). The values of objective functions
which
U (see Figure 4). The values of objective functions f ( G1 ), f ( G2 ), f ( G3 ) and
, f (formed
G3 ) and a frectangle
(G4 ) were calculated. The interior point that had the minimum of the objective function f ( G4was denoted as temp point. The rectangle ) were calculated. The interior point that U
had
the minimum of the objective
denoted
, G 2 ' , G3was
G1 'function
' , and G4 ' ' with the new interior points
as temp
point.
The
rectangle
U
0
with
the
new
interior
points
G
0
,
G
0
,
G
0
,
and
G
0
was
retained
2
3of the rectangle 4
was retained according to the temp point. After each iteration, 1the area U ' was according
to the
temp point.
After
iteration,
the area
the rectangle
U 04). was
gradually decreased, until the each
desired precision was of
obtained (Figure A gradually
four‐step decreased,
iterative untilalgorithm given below was applied to implement the CTOS & LSR. the desired precision was obtained (Figure 4). A four-step iterative algorithm given below was
applied to implement the CTOS & LSR.
Forests 2016, 7, 194
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Forests 2016, 7, 194 b2
6 of 20 P3
P4
G3
b2
P3
G3
G4
G1
U
G1
a2 P
1
a1
G2 G2
G4
U
G2 G2
P1
P2
a2
P2
(1)
P4
a1
b1
(2)
b1
Figure 4. Diagram of the combination of constrained two‐dimensional optimum seeking method and Figure
4. Diagram of the combination of constrained two-dimensional optimum seeking method and
least square regression (CTOS & LSR), P1 , P2 , P3 , and P4 are four corners of the rectangle U least square regression (CTOS & LSR), P1 , P2 , P3 , and P4 are four corners of the rectangle U determined
G4 2 , G
3 , and points
by determined by the feasible region of the estimated two‐dimensional variables, the feasible region of the estimated two-dimensional variables, G1 , G2 , G3 , andG1G, 4Gare
interior
determined by the golden section method, are interior points of P2 ' , corners
P3 ' , and ofP4the
' are four 1 ' , four
of U
determined by theUgolden
section method, P1 0, P2 0, P3 0, and P4 0Pare
rectangle
0

obtained by one iteration calculation of CTOS & LSR; , corners of the rectangle G
'
G
'
' , U 0
U
U obtained by one iteration calculation of CTOS & LSR; G1 0, G2 0, G3 0, and G4 0 are interior
points
1
2 , G3 of

determined by the golden section method. and G4 ' are interior points of U method.
determined
by the golden section
The feasible region variable 1 was feasible region of variable 2 was 1 ) and StepStep 1: 1: The
feasible
region
of of variable
1 was
(a1 ,( ba11), band
the the feasible
region
of variable
2 was
(a2 , b2 ).
( a2 ,four
formed by the overlapped area were represented b2 ). The four corners of a rectangle U by
The
corners of a rectangle U formed
the overlapped area were represented by P1 , P2 ,
interior P1 , P
P42 ,, respectively.
P3 , and P4 , respectively. G1 , G
G3 , and G4 U , four Pby In rectangleIn U,rectangle four interior
points
G1 , Gpoints G24, were
obtained
3 , and
2 , G3 , and
were obtained from the intersections of the following four lines: from
the intersections of the following four lines:
x1 x1a1= a0.382(
b1 (ba11 )−, a1 ),
1 + 0.382
(3) (3)
1 + 0.618
x2 x2a1=a0.618(
b1 (ba11 )−, a1 ),
(4) (4)
y1 = a2 + 0.382(b2 − a2 ),
y1  a2  0.382(b2  a2 ) , (5) y  a  0.618(b  a ) , (6) y2 = a2 + 0.618(b2 − a2 ),
(5)
(6)
2
2
The distance from the point P1 with
coordinates
(a21 , a2 )2 to the point P4 with coordinates (b1 , b2 )
was calculated
by the following formula:
The distance from the point P1 with coordinates ( a1 , a2 ) to the point P4 with coordinates ( b1 , b2 ) was calculated by the following formula: D =k p1 − p4 k2 =
q
( a1 − a2 )2 + (b1 − b2 )2 .
D || p1  p4 ||2  (a1  a2 )2  (b1  b2 )2 . Step 2:
(7) (7)
θ in Equation (1) was estimated at each interior point with known coordinates by LSR, the
Step 2: θ in Equation (1) was estimated at each interior point with known coordinates by LSR, the objective functions f ( G1 ), f ( G2 ), f ( G3 ) and f ( G4 ) were computed, and the interior point that
objective functions computed, and the interior point f (G1 ) , ffunction
(G2 ) , f (G
f (G
had
the minimum
objective
value
was
denoted
temp point.
3 ) and 4 ) were as
Ifthat had the minimum objective function value was denoted as temp point. f ( G1 ) was the minimum, then the rectangle with P1 G4 as the secondary diagonal was
Step 3: If was minimum, then the the
rectangle with the the
secondary diagonal was f (G1 ) if
P1GG4 3 as reserved;
f ( Gthe the minimum,
rectangle
with
P2 as
secondary
diagonal
was
2 ) was
Step 3:
Step 4:
was the minimum, the rectangle with reserved; if (G32)) was
reserved;
if f (f G
the minimum, the rectangle withGP33PG
2 as the secondary diagonal was 2 as the secondary diagonal was
reserved;
if f (f G
the minimum, the rectangle withP3GG12P as the secondary diagonal was reserved; if was the minimum, the rectangle with (G43)) was
4 as the secondary diagonal was
reserved.
The
four
corners
of
the
new
rectangle
U
0
were
denoted
as P1 0, P2 0, P3 0, and P4 0,
reserved; if was the minimum, the rectangle with as the secondary diagonal was f (G
)
G
P
4
1 4
respectively.
U
0
,
the
four
interior
pointsG
0
,
G
0
,
G
0
,
and
G
0
in
the
new
also were
2 U '3 were denoted as 1
4
reserved. The four corners of the new rectangle P1 ' , Prectangle
2 ' , P3 ' , and P4 ' , 0
determined
byUthe
golden
distance
D rectangle =k P1 0 −also P4 0 k2 .
four section
interior method.
points G1 Computed
and G 4 'would
in the be
new respectively. ' , G 2 ' , G3 ' , ' , the Steps
2
and
3
were
repeated
until
a
desired
precision
was
obtained,
for
example,
were determined by the golden section method. Computed distance would be D  || P1 ' P4 ' ||2
0
|D
. |< 0.00000001 . The final α̂1 and α̂2 were obtained, and then β̂1 − β̂4 were estimated
by LSR.
Forests 2016, 7, 194
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The CTOS & LSR with the four-step iterative algorithm for estimating in Equation (2) was
implemented in VB version 6.0, see Appendix A. It should be noted that, to simplify the LSR calculation,
the model parameters β1 − β4 in each iteration after giving α̂1 and α̂2 were estimated by transforming
Equation (1) into the following linear form:
y = β1 x1 + β2 x2 + β3 x3 + β4 x4 + ε
(8)
where, x1 = h/H − 1, x2 = (h/H )2 − 1, x3 = (α̂1 − h/H )2 I1 , x4 = (α̂2 − h/H )2 I2 , y = (d/D )2 , and
the corresponding indicator variables I1 and I1 are given by
I1 =
|(α̂1 − h/H )|+(α̂1 − h/H )
2(α̂1 − h/H )
(9)
I2 =
|(α̂2 − h/H )|+(α̂2 − h/H )
2(α̂2 − h/H )
(10)
Therefore, Equation (8) was a typical linear model, which was easily solved by LSR.
2.4. ULSR Method
The method ULSR based on the minimization of the residual sum of squares has been commonly
used to estimate parameters in forest growth and yield models. Equation (1) was difficult to solve
directly using ULSR due to the unknown parameters α1 and α2 in both indicator variables I1 and I2 .
Based on Equations (9) and (10), we transformed Equation (1) to the following form:
y = β1 ( x − 1) + β2 ( x2 − 1) + β3 (α1 − x )2
|(α1 − x )|+(α1 − x )
2 ( α1 − x )
α2 − x )
+ β4 (α2 − x )2 |(α2 −2(xα)|+(
+ε
−x)
2
(11)
This transformed Equation (11) is a general nonlinear model and could be directly estimated
by ULSR.
2.5. Taper Equation Evaluation
With application of Equation (1) the individual tree-level equation using sub-dataset from nine
isolated dominant or codominant trees (Figure 2) and the population average-level equation using the
full dataset of 120 trees (Table 1) were developed using both ULSR and CTOS & LSR. The predictive
ability of Equation (1) at the individual tree-level obtained from the two methods was assessed
using the coefficient of determination (R2 ) and the residual sum of squares (RSS) statistics that are
accounted for in Equations (12) and (13). The predictive ability of Equation (1) at the population
average-level was assessed using R2 , RSS, the mean of prediction errors (e) (Equation(14)), the variance
of prediction errors (σ2 ) (Equation (15)), and the root mean square error (δ) (Equation (16)) by a defined
cross-validation [38,39].
2
∑ (yi − ŷi )
2
R = 1−
(12)
2
∑ ( yi − y )
∑ (yi − ŷi )2
(13)
∑(yi − ŷi )/N
(14)
RSS =
e=
σ2 =
∑ (yi − ŷi )2 /( N − 1)
δ=
p
e2 + σ2
(15)
(16)
where yi and ŷi are the observed and predicted dependent for the ith observation; y is the mean of
the observed dependent, N is the total number of observations; e and σ2 are the mean and variance of
prediction errors, respectively; and δ is the root mean square error that combines the mean bias and
the variation of the residuals and was used as the primary criterion for taper equation evaluations [40].
Forests 2016, 7, 194
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In this study, the cross-validation meant that the full dataset was randomly divided into 10
sub-datasets, each having 12 trees and Equation (1) was fitted 10 times. Each time, one sub-dataset
(i.e., a total of 12 trees) was removed from the full dataset and the remaining nine sub-datasets
(i.e., a total of 108 trees) together were used to fit the equation using both methods of ULSR and
CTOS & LSR. The equations obtained were then employed to make tree taper predictions for the
removed sub-dataset and the differences between the estimated and observed values were calculated.
After all the sub-datasets yielded the predicted values of tree taper, the statistics (R2 , RSS, e, σ2 and
δ) of the differences between the estimated and observed values were calculated. For details of the
cross-validation approach, readers can refer to the studies by Vanclay [38] and Arlot and Celisse [39].
Moreover, all calculations were carried out using ForStat 2.2 version [41].
3. Results
3.1. Taper Equation at Individual Tree-Level
Table 2 shows the fitting results including parameter estimates and statistics for nine individual
trees using both ULSR and CTOS & LSR. The fitting of Equation (1) using both ULSR and CTOS & LSR
for each tree reached convergence with meaningful parameter estimates. The fitting of Equation (1)
using the ULSR did not reach convergence with one-time starting values. Thus, different starting
values were needed to make the equation convergence with the global minimum. However, with
CTOS & LSR, Equation (1) easily converged for all nine individual trees. The differences of some
parameter estimates of Equation (1) between ULSR and CTOS & LSR for all trees were significant at a
risk level of α = 0.05. For example, for E. fordii, the estimate of the joint point parameter α1 = 0.2431
from ULSR was almost four times larger than that (α1 = 0.0603) from CTOS & LSR, and the estimate
of joint point parameter α2 = 0.2818 from ULSR was almost one third of the corresponding parameter
estimate (α1 = 0.8665) from CTOS & LSR.
A t-test showed that the mean biases of all trees (Table 2) for the two methods were not significantly
different from zero (p > 0.05). The coefficients of determination R2 of each tree for the two methods
were very high. For many trees, such as C. hystrix No. 1 and No. 2, E. fordii No. 3, T. grandis
No. 2 and No. 3, the two methods had identical fitting precisions. However, for several trees, such as
E. fordii No. 2 and T. grandis No. 1, the RSS and R2 from the CTOS & LSR method were slightly better
than those of the corresponding trees obtained from ULSR. Especially for E. fordii No. 2, the value of
RSS from CTOS & LSR was 40% smaller than that from ULSR. These results suggested that the fitting
accuracy of Equation (1) at the individual tree-level obtained from CTOS & LSR was slightly higher
than that of the corresponding equation obtained from ULSR for those trees whose taper curve has
more variation.
For testing CTOS & LSR, the residuals of Equation (1) with the whole feasible ranges (0 < a1 <
a2 < 1) of the two joint point parameters α1 and α2 by step size 0.01 and satisfying the constrained
conditions α1 < α2 were calculated for the nine individual trees. The residual distribution of each
tree is displayed in Figure 5. The red points in Figure 5 denote the corresponding final solutions of
Equation (1) obtained by CTOS & LSR for each individual tree. The points with the minimum RSS
of Equation (1) with the constrained 0 < a1 < a2 < 1 were almost identical to the corresponding red
points that were determined by CTOS & LSR for the nine individual trees. This indicated that the
CTOS & LSR proposed in this study could minimize the RSS of Equation (1) more efficiently.
Forests 2016, 7, 194
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Table 2. Parameter estimates and statistics of Equation (1) fitted by a combined method of constrained two-dimensional optimum seeking and least square regression
(CTOS & LSR) and the unconstrained least square regression (ULSR) for nine individual trees (three trees from each species of C. hystrix, E. fordii, and T. grandis),
α1 − α2 , joint point parameters; β1 − β4 , model parameters; RSS, residual sum of square; R2 , coefficient of determination.
Tree Species
C. hystrix
E. fordii
T. grandi
No.
Method
ff1
ff2
fi1
fi2
fi3
fi4
RSS
R2
(1)
CTOS & LSR
ULSR
0.0579
0.0585
0.3937
0.3937
−2.8796
−2.8791
1.3293
1.329
139.5926
−3.3658
−3.368
129.354
0.0230
0.0230
0.991
0.991
(2)
CTOS & LSR
ULSR
0.0352
0.0356
0.5093
0.5092
−3.2136
−3.213
1.6851
1.6847
182.6908
173.213
−1.7546
−1.7545
0.0090
0.0090
0.996
0.996
(3)
CTOS & LSR
ULSR
0.2292
0.2812
0.3783
0.3407
−2.1126
−2.1712
0.8538
0.8912
25.3279
32.6352
−7.4583
−22.27
0.0074
0.0066
0.997
0.997
(1)
CTOS & LSR
ULSR
0.1595
0.1649
0.7361
0.7675
−4.7264
−5.9143
2.3894
3.0609
27.0427
21.8090
−2.7965
−3.4446
0.0204
0.0206
0.989
0.989
(2)
CTOS & LSR
ULSR
0.2431
0.0603
0.2818
0.8665
−2.1716
2.0566
0.9591
−1.4212
93.5096
92.6192
−53.8856
2.3332
0.0299
0.0499
0.985
0.9774
(3)
CTOS & LSR
ULSR
0.0340
0.0342
0.3631
0.3695
−2.6455
−2.6554
1.3478
1.3541
886.3202
844.7019
−1.7084
−1.667
0.0029
0.0029
0.999
0.999
(1)
CTOS & LSR
ULSR
0.3996
0.4916
0.5837
0.5549
−4.4012
−4.0975
2.3442
2.1651
7.9341
9.0758
−5.2571
−9.5458
0.0179
0.0190
0.990
0.990
(2)
CTOS & LSR
ULSR
0.1385
0.1403
0.6910
0.8530
−2.8939
−8.4821
1.4189
4.5000
33.1479
27.9491
−1.2589
−4.2567
0.0026
0.0026
0.999
0.999
(3)
CTOS & LSR
ULSR
0.0325
0.0328
0.5609
0.5611
−3.4995
−3.5011
1.7276
1.7286
558.3056
532.4962
−2.2966
−2.2968
0.0134
0.0134
0.995
0.995
distribution of each tree is displayed in Figure 5. The red points in Figure 5 denote the corresponding final solutions of Equation (1) obtained by CTOS & LSR for each individual tree. The points with the minimum RSS of Equation (1) with the constrained 0  a1  a2  1 were almost identical to the corresponding red points that were determined by CTOS & LSR for the nine individual trees. This Forests
2016, 7, 194
indicated that the CTOS & LSR proposed in this study could minimize the RSS of Equation 10
(1) of 20
more efficiently. C. hystrix No. 1 C. hystrix No. 2 C. hystrix No. 3 E. fordii No. 2 E. fordii No. 3 T. grandi No. 2 T. grandi No. 3 E. fordii No. 1 T. grandi No. 1 Figure 5. The residual distribution of Equation (1) with the whole feasible range ( 0      1 ) of Figure 5. The residual distribution of Equation (1) with the whole feasible range (0 < 1α1 <2 α2 < 1) of
joint points 1 and  2 by step size 0.01 and satisfying the constrained conditions 1   2 calculated joint points α1 and α2 by step size 0.01 and satisfying the constrained conditions α1 < α2 calculated by
the combination the constrained two‐dimensional optimum seeking method the least theby combination
of theof constrained
two-dimensional
optimum
seeking
method
and and the least
square
square regression (CTOS & LSR) for the nine individual trees. The red points denote the regression (CTOS & LSR) for the nine individual trees. The red points denote the corresponding final
corresponding final solutions of Equation (1) obtained by CTOS & LSR for each individual tree. solutions of Equation (1) obtained by CTOS & LSR for each individual tree.
3.2. Taper Equation at Population Average‐Level 3.2. Taper Equation at Population Average-Level
The parameter estimates of Equation (1) obtained from both ULSR and CTOS & LSR based on The parameter estimates of Equation (1) obtained from both ULSR and CTOS & LSR based on the
the full dataset (Table 1) were listed in Table 3. Equation (1) for each tree species reached convergence fullwith dataset
(Table 1)parameter were listed
in Tableusing 3. Equation
(1) for each
reached
with
meaningful estimates either method. As tree
with species
Equation (1) at convergence
the individual meaningful
estimates
usingwere either
method.for AsULSR with Equation
(1) at the (1) individual
tree-level,
tree‐level, parameter
different starting values required to make Equation converge. The different
starting values were required for ULSR to make Equation (1) converge. The parameter
parameter estimates of Equation (1) with ULSR and CTOS & LSR for each species were not identical. estimates of Equation (1) with ULSR and CTOS & LSR for each species were not identical. Especially
for C. hystrix, the value of β3 estimated by ULSR was 7% larger than that estimated by CTOS & LSR.
Based on cross-validation, the assessment results of five prediction statistics [Equations (12) and
(16)] of Equation (1) using both ULSR and CTOS & LSR for the three tree species are listed in Table 4.
The mean prediction errors e of Equation (1) for each tree species using either method were very small.
Equation (1) for each tree species using both ULSR and CTOS & LSR slightly over-predicted the d2 /D2 .
The coefficient of determination R2 and RSS of Equation (1) for each tree species from ULSR and
CTOS & LSR were correspondingly identical. The prediction accuracies of Equation (1) for each tree
species from CTOS & LSR were much higher than those of Equation (1) when the d2 /D2 was estimated
using from ULSR. For example, the values of σ2 and δ of Equation (1) from CTOS & LSR for each tree
species were smaller than those of Equation (1) from ULSR. Especially for C. hystrix, the variance of
Forests 2016, 7, 194
11 of 20
prediction errors σ2 from CTOS & LSR was 8.34% smaller than those from ULSR, and the value of root
mean square error δ from CTOS & LSR was 5% smaller than those from ULSR. This suggested that
the prediction ability of Equation (1) using CTOS & LSR was more powerful than that of Equation (1)
using ULSR.
Table 3. Parameter estimates of Equation (1) obtained from both ULSR and CTOS & LSR based on
the full data-set from three species (C. hystrix, E. fordii, and T. grandi), α1 − α2 , joint point parameters;
β1 − β4 , model parameters.
Tree Species
Method
α1
α2
β1
β2
C. hystrix
ULSR
CTOS & LSR
0.0481
0.0474
0.6479
0.6479
−3.6251
−3.624
1.7049
1.7043
158.3459 −2.1074
169.9877 −2.1063
E. fordii
ULSR
CTOS & LSR
0.0444
0.0440
0.7190
0.6910
−2.5992
−2.4256
1.1148
1.0154
257.2861 −0.7817
274.0969 −0.6923
T. grandis
ULSR
CTOS & LSR
0.0903
0.0890
0.6993
0.6910
−2.5923
−2.5193
1.1282
1.0861
51.1553
52.6617
β3
β4
−0.9210
−0.8752
Table 4.
The statistics of Equation (1) fitted by the combination method of constrained
two-dimensional optimum seeking and least square regression (CTOS & LSR) and the unconstrained
least square regression (ULSR) for three trees species (C. hystrix, E. fordii, and T. grandis) based on
cross-validation approach.
Tree Species
Method
RSS
R2
e
σ2
δ
C. hystrix
ULSR
CTOS & LSR
3.404
3.404
0.959
0.959
−0.2518
−0.1814
2.2898
2.0989
1.5340
1.4601
E. fordii
ULSR
CTOS & LSR
4.401
4.401
0.953
0.953
−0.1568
−0.1611
2.1878
2.1753
1.4874
1.4837
T. grandis
ULSR
CTOS & LSR
3.479
3.479
0.962
0.962
−0.2211
−0.1870
3.7328
3.5452
1.9446
1.8921
RSS, residual sum of square; R2 , coefficient of determination; e, mean of prediction errors; σ2 , variance of
prediction errors; δ, root mean square error.
The residuals of Equation (1) for each tree species using both ULSR and CTOS & LSR were plotted
against the observed values (Figure 6). The results showed that Equation (1) for each tree species using
either ULSR or CTOS & LSR did not lead to any trend of heteroscedasticity, which further indicated
that Equation (1) was a better segmented taper equation for the prediction of tree taper.
Forests 2016, 7, 194
12 of 20
Forests 2016, 7, 194 12 of 20 10
10
ULSR
CTOS&LSR
C. hystrix
5
5
0
0
-5
-5
-10
-10
0
10
10
20
30
40
50
0
10
20
30
40
50
10
ULSR
E. fordii
5
Residual
C. hystrix
CTOS&LSR E. fordii
5
0
0
-5
-5
-10
-10
0
10
20
30
40
50
0
10
10
20
30
40
50
10
ULSR
5
CTOS&LSR
T. grandis
5
0
0
-5
-5
-10
T. grandis
-10
0
10
20
30
40
50
d
0
10
20
30
40
50
Figure
6. Residuals of the Equation (1) for the three tree species C. hystrix, E. fordii. and T. grandi
Figure 6. Residuals of the Equation (1) for the three tree species C. hystrix, E. fordii. and T. grandi using using
both
combination
method
of constrained
two-dimensional
optimum
seeking
least
square
both the the
combination method of constrained two‐dimensional optimum seeking and and
least square regression
(CTOS & LSR) and the unconstrained least square regression (ULSR) plotted against the
regression (CTOS & LSR) and the unconstrained least square regression (ULSR) plotted against the observed
values based on cross-validation.
observed values based on cross‐validation. 4. 4. Discussion Discussion
Equation (1) is able to describe the shape of a tree. The bottom trunk section, which includes the Equation
(1) is able to describe the shape of a tree. The bottom trunk section, which includes the
butt, is modeled as a frustum of a neiloidal solid, the middle of the trunk is assumed to take the shape butt, is modeled as a frustum of a neiloidal solid, the middle of the trunk is assumed to take the shape
of a frustum of a parabolodial solid, and the top portion is assumed to be conoid [13]. In this system, of a frustum of a parabolodial solid, and the top portion is assumed to be conoid [13]. In this system,
three additive equations were grafted together by incorporating two joint points between the three three additive equations were grafted together by incorporating two joint points between the three
segments. The joint point parameters 1 and  2 with the constrained condition 0  1   2  1 in segments. The joint point parameters α1 and α2 with the constrained condition 0 < α1 < α2 < 1 in
Equation (1) were previously commonly estimated by ULSR and SE, which made the results of the Equation (1) were previously commonly estimated by ULSR and SE, which made the results of the
equation uncertain. We proposed the CTOS & LSR method to estimate the parameters of Equation equation uncertain. We proposed the CTOS & LSR method to estimate the parameters of Equation (1).
(1). It was based on the constrained two‐dimensional optimum seeking algorithm with Hua Luogeng’s It was
based on the constrained two-dimensional optimum seeking algorithm with Hua Luogeng’s
theory [36]. The CTOS & LSR method was evaluated using the stem analysis data sets of three tropical theory
[36]. The CTOS & LSR method was evaluated using the stem analysis data sets of three
precious tree species C. hystrix, E. fordii, and T. grandis in the southwest of China. Our results (Tables 2 tropical precious tree species C. hystrix, E. fordii, and T. grandis in the southwest of China. Our results
Forests 2016, 7, 194
13 of 20
Forests 2016, 7, 194 13 of 20 (Tables
2 and
4) show
thatmethod this method
is more
reliable
ULSR
for estimating
parameters
in the
and 4) show that this is more reliable than than
ULSR for estimating the the
parameters in the segmented
taper equation.
segmented taper equation. The literature show that ULSR is a widely used method for parameter estimation [15,24,28–32]. The
literature show that ULSR is a widely used method for parameter estimation [15,24,28–32].
This method was also applied in the CTOS & LSR. For each iterative calculation in CTOS & LSR, the This
method was also applied in the CTOS & LSR. For each iterative calculation in CTOS & LSR, the
CTOS was first used estimate parameters
parameters α
parameters were 11 and CTOS
was first used toto estimate
and α22, after , afterthe thetwo joint point two joint point
parameters were
determined, other remaining parameters in Equation (1) were estimated by LSR without constraint. determined,
other remaining parameters in Equation (1) were estimated by LSR without constraint.
parameter estimates of Equation (1) the
for the methods obtained by minimizing the TheThe parameter
estimates
of Equation
(1) for
twotwo methods
werewere obtained
by minimizing
the similar
similar objective function (residual sum of squares). However, for ULSR, the estimates of parameters objective function (residual sum of squares). However, for ULSR, the estimates of parameters may fall
outmay fall out of the feasible range and make the results meaningless. For example, for the individual of the feasible range and make the results meaningless. For example, for the individual tree of
, of (Figure
E. fordii (Figure 7), the fitting accuracies of Equation RSS  0.0460
E. tree fordii
7), the fitting
accuracies of Equation
(1)(1) obtained from ULSR obtained from ULSR ((RSS
= 0.0460,
2
2
), and CTOS & LSR (
, ) were almost similar and very high R

0.9791
R

0.9823
RSS

0.0391
R2 = 0.9791), and CTOS & LSR (RSS = 0.0391, R2 = 0.982) were almost similar and very high
from isULSR is ˆ 2 (the
(Figure 7). However, the estimate of 2ULSR
1.3130 (the other estimate was (Figure
7). However,
the estimate
of α2 from
α̂2 = 1.3130
other estimate was α̂1 = 0.1915),
), which was larger than one and out of the feasible range. For CTOS & LSR, the estimates ˆ 1 was
0.1915
which
larger
than one and out of the feasible range. For CTOS & LSR, the estimates of α1 and α2
of 0.2692
1 and and
2 were 0.2692 and 0.6943; they were within the feasible range. Figure 7 also shows that were
0.6943; they were within the feasible range. Figure 7 also shows that the predicted
taper
curve
from
ULSR almost tends to zero when the relative height h/H is just 0.92,
conflicts
is just 0.92, the predicted taper curve from ULSR almost tends to zero when the relative height h / Hwhich
with
the assumption that the relative outside bark diameter d/D should be equal
to zero at the relative
which conflicts with the assumption that the relative outside bark diameter d / D should be equal to height
h/H = 1. This situation
occurred in our preliminary analyses.
zero at the relative height h / Halways
 1 . This situation always occurred in our preliminary analyses. 1.6
Observation
ULSR
CTOS&LSR
1.4
1.2
d/D
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
h/H
0.8
1.0
1.2
Figure 7. Observed relative outside bark diameters at different relative height location (black line) for Figure
7. Observed relative outside bark diameters at different relative height location (black line)
an individual taper curves from the Equation (1) (1)
using both the the
for an individualtree treeof ofE. E.fordii. fordii.The Thepredicted predicted
taper
curves
from
the
Equation
using
both
combination method of constrained two‐dimensional optimum seeking and least square regression combination method of constrained two-dimensional optimum seeking and least square regression
(CTOS & LSR) (red line) and the unconstrained least square regression (ULSR) (blue line) are shown. (CTOS
& LSR) (red line) and the unconstrained least square regression (ULSR) (blue line) are shown.
The overall prediction accuracies of Equation (1) at individual tree‐levels and population The overall prediction accuracies of Equation (1) at individual tree-levels and population
average‐levels from CTOS & LSR were much larger than those from ULSR (Tables 2 and 4). This is average-levels
from CTOS & LSR were much larger than those from ULSR (Tables 2 and 4). This is
due to the fact that the optimum solutions of Equation (1) obtained from CTOS & LSR were closer to dueactual values than these obtained from ULSR (Figure 5). In addition, the ULSR approach does not to the fact that the optimum solutions of Equation (1) obtained from CTOS & LSR were closer to
actual
values than these obtained from ULSR (Figure 5). In addition, the ULSR approach does not
always converge for parameter estimation and this problem becomes more serious while fitting the always
converge for parameter estimation and this problem becomes more serious while fitting the
taper equation at the individual tree‐level. This is because the convergence of ULSR based on the iteration optimization algorithm, i.e., Gauss‐Newton, depends largely on the given initial values of taper
equation at the individual tree-level. This is because the convergence of ULSR based on the
the parameters. In contrast, CTOS & LSR does not have a divergence problem. We also tried using iteration
optimization algorithm, i.e., Gauss-Newton, depends largely on the given initial values of the
parameters. In contrast, CTOS & LSR does not have a divergence problem. We also tried using other
Forests 2016, 7, 194
14 of 20
Forests 2016, 7, 194 14 of 20 other existing constrained optimization method e.g., simplex method and constrained Newton existing constrained optimization method e.g., simplex method and constrained Newton Raphson
Raphson method, to estimate the parameters in Equation (1), but these methods failed to converge method, to estimate the parameters in Equation (1), but these methods failed to converge for some
for some individual trees or tree species. Especially for the simplex method, the computational speed individual trees or tree species. Especially for the simplex method, the computational speed was also
was also very slow. Besides forestry, CTOS & LSR could be also applied in agricultural engineering, very
slow. Besides
forestry,
& LSR
could selection be also applied
in agricultural
engineering,
industry or other fields CTOS
requiring optimal of parameters. It is to be noted industry
that the or
other
fields
requiring
optimal
selection
of
parameters.
It
is
to
be
noted
that
the
CTOS
&
LSR
method
CTOS & LSR method only works on unimodal models. For multi‐modal functions, there is no only
works
on
unimodal
models.
For
multi-modal
functions,
there
is
no
guarantee
that
the
solutions
guarantee that the solutions of CTOS & LSR are the global optimum. We will focus on this limitation of
CTOS
& LSR are the global optimum. We will focus on this limitation of the method in our future work.
of the method in our future work. Equation
(1) with the estimates of parameters obtained from CTOS & LSR (Table 3) could be
Equation (1) with the estimates of parameters obtained from CTOS & LSR (Table 3) could be used
to demonstrate
differences
of tree
tapertaper among
threethree species
C. hystrix,
E. fordii,
T. grandis.
used to demonstrate differences of tree among species C. hystrix, E. and
fordii, and TheT. grandis. The prediction accuracy with Equation (1) at the population average‐level for each tree prediction accuracy with Equation (1) at the population average-level for each tree species is
species is high (Figure 8). In addition, there are apparent differences among the tree taper of the three high
(Figure 8). In addition, there are apparent differences among the tree taper of the three tropical
tropical precious tree species. For example, when the range of / H was from 0.00 to 0.03, the values precious
tree species. For example, when the range of h/H washfrom
0.00 to 0.03, the values of d/D for
of d / Dwere
for T. grandis were much larger than those for the other two species (C. hystrix and E. fordii), T. grandis
much larger than those for the other two species (C. hystrix and E. fordii), which have
which have almost the same . When the range of was from 0.03 to 0.7, the values of d/D
h / H0.03
/D almost
the same d/D. When the
range
of h/H was from
to 0.7, the values of d/D for C.d hystrix
for C. hystrix was the largest, following by E. fordii and T. grandis. When is larger than 0.8, the h
/
H
was the largest, following by E. fordii and T. grandis. When h/H is larger than 0.8, the three species
three species have the same tree taper. Equation (1) also could be used to describe differences in tree have the same tree taper. Equation (1) also could be used to describe differences in tree taper between
taper between individual trees. individual trees.
1.4
C. hystrix
E. fordii
T. grandis
1.2
d/D
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
h/H
0.8
1.0
1.2
Figure 8. The predicted taper curves from Equation (1) using the combined method of constrained Figure
8. The predicted taper curves from Equation (1) using the combined method of constrained
two‐dimensional optimum seeking and least square regression (CTOS & LSR) for the three tree species two-dimensional
optimum seeking and least square regression (CTOS & LSR) for the three tree species
C. hystrix (red line), E. fordii (black line), and T. grandi (blue line) at the population average level. C. hystrix
(red line), E. fordii (black line), and T. grandi (blue line) at the population average level.
Measuring tree diameter and height may be subject to errors, even though stand and tree Measuring tree diameter and height may be subject to errors, even though stand and tree variables
variables are commonly assumed to be measured without error [42,43]. The measurement errors aremade commonly
assumed
be measured
without
[42,43].
The measurement
errors made
by field crew or to
faulty instrument or both error
might be substantial [42]. In all existing taper by
field
crew or faulty instrument or both might be substantial [42]. In all existing taper equations
equations including Equation (1), it is assumed that (i) the dependent variable is a random variable; including
Equation (1), it is assumed that (i) the dependent variable is a random variable; and (ii) the
and (ii) the independent variables are fixed and measured without errors. It is well known that the independent
variables
are fixed
and measured
without
errors.
It is wellof known
that the
violation
of the
violation of the second assumption may lead to biased estimation parameters and (or) of the second
assumption may lead to biased estimation of parameters and (or) of the standard errors of the
standard errors of the parameters, and consequently to misleading the hypothesis test [44,45]. If the predictor variables in Equation (1) are considered to have measurement a new parameter parameters,
and consequently
to misleading
the hypothesis
test [44,45]. Iferrors, the predictor
variables in
Equation (1) are considered to have measurement errors, a new parameter estimation approach needs
to be developed. We are in the process of developing such a method to deal with this problem.
Forests 2016, 7, 194
15 of 20
5. Conclusions
In this study, the CTOS & LSR was proposed as an improved method to estimate the parameters
in the Equation (1). This method was compared with ULSR for both individual tree-level equation
and the population average-level equation using data from three tree species including C. hystrix,
E. fordii, and T. grandis in the southwest of China. The differences between CTOS & LSR and ULSR
were found to be significant. The segmented taper equation estimated using CTOS & LSR resulted
in not only increased prediction accuracy, but also guaranteed the parameter estimates in a more
meaningful way. It is thus recommended that the CTOS & LSR should be a preferred choice for
this application.
Acknowledgments: We are grateful to Shouzheng Tang, Yuancai Lei, Yuanchang Lu and other researchers in the
Chinese Academy of Forestry for their valuable suggestions on this study. We also thank the Experimental Center
of Tropical Forestry, Chinese Academy of Forestry Sciences for providing friendly help during the data collection.
Financial support for this study was provided by the National High-tech R & D Program of China (863 Program)
(No. 2012AA102002), the Basic Scientific Research Business of Central Public Research Institutes (No. IFRIT2013),
the Forestry Public Welfare Scientific Research Project of China (No. 201404417) and the National Natural Science
Foundations of China (Nos. 31300534, 31570628, 31470641).
Author Contributions: L.F. and L.P. conceived the study. L. F. L.P., Y.M. amd X.S. performed the analysis and
wrote the initial draft of the paper. All authors contributed to interpreting results and improvement of the paper.
Conflicts of Interest: The authors declare that they have no conflict of interest.
Appendix
A VB program for estimating the Max and Burkhart ([13] 1976) taper Equation (1) using CTOS &
LSR with the four-step iterative algorithm is shown here.
Function Do_Function_Two_dimensional_optimization_selection(Independ_Col() As Long,
dePendent_Col() As Long, Independ_Name() As String, dePendent_Name() As String,
a_ValueRegion() As Double, b_ValueRegion() As Double, startPoint As Long, EndPoint
As Long)
Dim Independ_value() As Double, dePendent_value() As Double, n As Long, i As Long,
j As Long, xyData() As Double, Para_value() As Double, Mat_result() As Variant
n = EndPoint
− startPoint + 1
ReDim Independ_value(n, UBound(Independ_Col))
ReDim dePendent_value(n, UBound(dePendent_Col))
For i = 1 To n
For j = 1 To UBound(Independ_Col)
Independ_value(i, j) = frmData.DataTab1.TextMatrix(i + startPoint
Independ_Col(j))
− 1,
Next j
For j = 1 To UBound(dePendent_Col)
− 1, dePendent_Col(j)) = "" Then
frmData.DataTab1.TextMatrix(i + startPoint − 1, dePendent_Col(j)) = "00"
If frmData.DataTab1.TextMatrix(i + startPoint
End If
dePendent_value(i, j) = frmData.DataTab1.TextMatrix(i + startPoint
dePendent_Col(j))
Next j
Next i
ReDim xyData(1 To n, UBound(Independ_Col) + UBound(dePendent_Col))
For i = 1 To n
For j = 1 To UBound(Independ_Col)
xyData(i, j) = Independ_value(i, j)
Next j
− 1,
Forests 2016, 7, 194
16 of 20
Next i
For i = 1 To n
For j = 1 To UBound(dePendent_Col)
xyData(i, j + UBound(Independ_Col)) = dePendent_value(i, j)
Next j
Next i
ReturnVal = two_dimensional_optimization_selection_Tang(xyData(), a_ValueRegion(),
b_ValueRegion(), Para_value())
ReDim Mat_result(1, 7)
Mat_result(0, 1) = "a1" : Mat_result(0, 2) = "a2" : Mat_result(0, 3) = "b1"
Mat_result(0, 4) = "b2" : Mat_result(0, 5) = "b3" : Mat_result(0, 6) = "b4"
Mat_result(0, 7) = "E2" : Mat_result(1, 1) = ReturnVal(1) :
Mat_result(1, 2) = ReturnVal(2)
Mat_result(1, 3) = Para_value(1) : Mat_result(1, 4) = Para_value(2)
Mat_result(1, 5) = Para_value(3) : Mat_result(1, 6) = Para_value(4)
Mat_result(1, 7) = ReturnVal(0)
With frmOutput.Text1
Call TextPrint(Mat_result, PrintFormat0, , , 0)
End With
Call Do_WriteOutputTxt
Exit Function
Function two_dimensional_optimization_selection_Tang(xyData()
As Double, a_ValueRegion() As Double, b_ValueRegion() As Double, Para_value() As Double)
Dim i As Integer, j As Integer, Sa1 As Double, Sb1 As Double, Sa2 As Double, Sb2 As Double,
x1 As Double, x2 As Double, Y1 As Double, y2 As Double, Nx1 As Double, Nx2 As Double,
Ny1 As Double, Ny2 As Double, r1 As Double, r2 As Double, R3 As Double, R4 As Double,
temp As Double, dis As Double, Best1 As Double, Best2 As Double, ER As Double,
reVal() As Double
Sx1 = a_ValueRegion(1)
Sx2 = a_ValueRegion(2) 'the feasible range for the first variable
Sy1 = b_ValueRegion(1)
Sy2 = b_ValueRegion(2) 'the feasible range for the second variable
x1 = Sx1
x2 = Sx2
Y1 = Sy1
y2 = Sy2
dis = ((x1
− x2)ˆ2 + (Y1 − y2)ˆ2)ˆ0.5
While (dis >0.0000000001)
−
−
Ny1 = Y1 + (y2 −
Ny2 = Y1 + (y2 −
×
×
Y1) ×
Y1) ×
Nx1 = x1 + (x2
x1)
0.382
Nx2 = x1 + (x2
x1)
0.618
0.382
0.618
r1 = ErrorC(Nx1, Ny2, xyData, Para_value)
r2 = ErrorC(Nx2, Ny2, xyData, Para_value)
R3 = ErrorC(Nx1, Ny1, xyData, Para_value)
R4 = ErrorC(Nx2, Ny1, xyData, Para_value)
If (r1 <r2) Then
temp = r1
Else
Forests 2016, 7, 194
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temp = r2
End If
If (R3 <temp) Then
temp = R3
End If
If (R4 <temp) Then
temp = R4
End If
If (temp = r1) Then
Y1 = Ny1
x2 = Nx2
Best1 = Nx1
Best2 = Ny2
ElseIf (temp = r2) Then
x1 = Nx1
Y1 = Ny1
Best1 = Nx2
Best2 = Ny2
ElseIf (temp = R3) Then
x2 = Nx2
y2 = Ny2
Best1 = Nx1
Best2 = Ny1
Else
x1 = Nx1
y2 = Ny2
Best1 = Nx2
Best2 = Ny1
End If
dis = ((x1
− x2)ˆ2 + (Y1 − y2)ˆ2)ˆ0.5
ER = ErrorC(Best1, Best2, xyData, Para_value)
ReDim reVal(2)
reVal(0) = ER
reVal(1) = Best1
reVal(2) = Best2
two_dimensional_optimization_selection_Tang = reVal
Exit Function
End Function
Function ErrorC(Best1 As Double, Best2 As Double, xyData() As Double, Para_value()
As Double)
Dim ReturnVal() As Double, i As Integer, j As Integer, iniParValues() As Double,
X() As Double, Y() As Double, n As Integer, x1() As Double, x2() As Double,x3() As Double,
x4() As Double, EstVal() As Variant, ResVal() As Variant, Sum_Res As Double, Rel() As Double
n = UBound(xyData, 1)
ReDim x1(n): ReDim x2(n): ReDim x3(n): ReDim x4(n)
For i = 1 To n
x1(i) = xyData(i, 1)
x2(i) = xyData(i, 1)ˆ2
− 1
− 1
If Best1 >= xyData(i, 1) Then
Forests 2016, 7, 194
x3(i) = (Best1
18 of 20
− xyData(i, 1))ˆ2/(Best1 − x1(i))
Else
x3(i) = 0
End If
If Best2 >= xyData(i, 1) Then
x4(i) = (Best2
− xyData(i, 1))ˆ2
Else
x4(i) = 0
End If
Next i
ReDim X(n, 4)
For i = 1 To n
X(i, 1) = x1(i)
X(i, 2) = x2(i)
X(i, 3) = x3(i)
X(i, 4) = x4(i)
Next i
ReDim Y(n)
For i = 1 To n
Y(i) = xyData(i, 2)
Next i
Rel = LinReg(X, Y)
ReDim Para_value(4)
For i = 1 To 4
Para_value(i) = Rel(i, 1)
Next i
ReDim EstVal(n)
ReDim ResVal(n)
Sum_Res = 0
For i = 1 To n
EstVal(i) = 0
For j = 1 To 4
EstVal(i) = EstVal(i) + Para_value(j)
× X(i, j)
Next j
ResVal(i) = (Y(i) − EstVal(i))ˆ2
Sum_Res = Sum_Res + ResVal(i)
Next i
ErrorC = Sum_Res
Exit Function
End Function
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© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access
article distributed under the terms and conditions of the Creative Commons Attribution
(CC-BY) license (http://creativecommons.org/licenses/by/4.0/).